the-game-theory-behind-stock-based-comp

If you have worked in a large company that issues stock to its employees, you might have observed that employees don't necessarily feel like "owners". As an employee, it is common to treat the stock price as some externality that is independent of your work.

It is paradoxical considering that the entire premise of stock based compensation is that employees do have an impact on the company's valuation and should be incentivized to increase it by compensating them with the company's shares.

In this post, I want to assume a simple model and illustrate the game theory behind this phenomenon.

Let's say a company called *ZikZok* has $n$ employees, and each employee is granted 1 share of *ZikZok*'s stock - meaning the company is 100% owned by its $n$ employees.

Now, each employee produces $v$ units of value for the company at $c$ units of cost, meaning the employee endures $c$ units of cost for producing $v$ units of value (this represents the time away from other activities, difficulty of the task, opportunity cost etc...). For their effort, they are compensated with a wage $w$ (in addition to their one share).

Adding $v$ units of value to the company makes it proportionately more valuable and the stock price rises $\frac{v}{n}$ (if market cap rises by $v$ and there are $n$ shares in circulation, the stock rises by $v/n$).

Now assume the employee's wage is $w$. If an employee works, they produce $v$ units of value and their payoff is: $w + \frac{v}{n} - c + \frac{kv}{n}$, where $k$ is the other number of employees that are producing value (i.e. working).

If the employee doesn't work, they are still getting paid, and are still capturing the value of the $k$ other workers but does not endure cost $c$. Their payoff is: $w + \frac{kv}{n}$.

The employee has an incentive to work if:

$\begin{aligned}& \text{payoff(work)} \geq \text{payoff(slacking off)} \\& \Leftrightarrow w + \frac{v}{n} - c + \frac{kv}{n} \geq w + \frac{kv}{n} \\& \Leftrightarrow \frac{v}{n} \geq c\end{aligned}$

As we can see, the incentive does not depend on the employee's wage nor the number other employees working, but rather whether the employee is able to capture a fraction of their value added, that is higher than the cost they endure.

The problem arises when $n$ increase: the fraction $\frac{v}{n}$ decreases, but the cost endured by the employee $c$ remains the same!

There is a point where the employee no longer has an incentive to work, which produces undesirable outcomes for the company.

**what's the critical number of employees, such that each of them has an incentive to play?**

Let's call $l$ the employee's leverage, such that $v=lc$. If $l \geq 1$, the employee can create more value than the cost they endure. Moreover, for the employee to have an incentive to work (recall $\frac{v}{n}\geq c$), the number of employees $n$ should not exceed $l$.

If $l < 1$, the employee's conversion of work to value added is inefficient and there is no number of employees such that they have an incentive to work.

**what's a better incentive scheme?**

The key insight from this model is that the employee's payoff should not decrease as the number of employees increases.

Instead of allocating 1 share per employee, let's imagine we allocate back a fraction $f\in [0,1]$ of the value produced $v$, such that the payoff for working is now $w + fv - c$ and the payoff for not working: $w$.

We can see that $f$ should be chosen such that $fv \geq c$.

This is the thinking behind commission based schemes such as sales teams or traders, which own a percentage of their profit and loss.

**Improving the model**

There are some adjustments we can make to make the model more realistic:

- we have considered the cost of working $c$ to be constant. But, I probably won't accept an offer that I would have accepted as a new grad. For instance, if I am especially productive, I would probably have alternative offers. Moreover, if my basic needs are taken care of, the cost of spending my time working as opposed to playing with my children should grow as well. To add more realism, we can make $c$ vary (non-linearly) with the wage $w$ and value produced $v$.
- we have treated individual workers and their value produced as completely independent. However, cooperation might increase value created at no added cost to the employee. There are a number of dynamics that can impact an individual's cost of working $c$ or value produced $v$.

**Closing thoughts**

When I started undertaking this exercise, I expected to observe something similar to the prisoner's dilemma with $n$ players. In that setting, **almost everybody** needs to play along in order to attain the maximal payoff, which becomes exponentially unlikely with $n$. What happens is called a *coordination failure*. I expected a similar "everybody needs to play along" scenario to come up in

The surprising part, to me, is that, unlike a classical prisoner's dilemma with $n$ players, the employee's incentive to work is not dependent on the probability that others are working. In a prisoner's dilemma with $n$ players, there is a coordination failure where the probability that each plays along should be unreasonably high, thus resulting in a nash equilibrium whose payoff is not maximal for any of the players.